ИСТИНА

In the papers Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka Journal of Mathematics, vol. 42(2) (2005); A. Kanel-Belov and M. Kontsevich, Automorphisms of Weyl algebras, Lett. Math. Phys. 74 (2005), 181-199; A. Kanel-Belov and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, arXiv: math/0512171v2, 2005 was constructed and discoursed homomorphism between (auto)endomorphisms of Weil algebra and polynomial symplectoendo(auto)morphisms. The construction dependant on the choice of infinitely large prime. We prove that for symplectoauthomorphisms in general case and symplectoendomorphisms in deformed case when [xi,\partialj]=ℏδij or {xi,yj}=ℏδij. My talk concerns recent progress made in the positive resolution of Kontsevich's conjecture, which states that, the procedure utilizes the following essential features. First, the Weyl algebra over an algebraically closed field of characteristic zero may be identified with a subalgebra in a certain reduced direct product (reduction modulo infinite prime) of Weyl algebras in positive characteristic -- a fact that allows one to use the theory of Azumaya algebras and is particularly helpful when eliminating the infinite series. Second, the lifting is performed via a direct homomorphism Aut Wn→Aut Pn which is an isomorphism of the tame subgroups (that such an isomorphism exists is known due to our prior work with Kontsevich) and effectively provides an inverse to it. Finally, the lifted automorphism is the limit (in formal power series topology) of a sequence of lifted tame symplectomorphisms; the fact that any polynomial symplectomorphism has a sequence of tame symplectomorphisms converging to it is our development of the work of D. Anick on approximation and is very recent. In order to make approximation work (this is not trivial at all because the ind-schemes are not reduced), we play with Plank constants and use singularity trick see coming to non-deformed case (it means that plank constant is not small parametr any more) is rather non-trivial and we don't understand how to proceed in endomorphism case. See [1] and [2] for details. (joint work with A. Elishev and J.-T. Yu)